Affine Hirsch foliations on 3-manifolds

TL;DR

This paper classifies affine Hirsch foliations on 3-manifolds, showing each admits up to two such foliations, constructs examples using braided links, and proves finiteness results related to strand numbers.

Contribution

It establishes the possible numbers of affine Hirsch foliations on 3-manifolds, constructs Hirsch manifolds via braided links, and proves finiteness of Hirsch manifolds with given strand number.

Findings

Every closed orientable 3-manifold admits 0, 1, or 2 affine Hirsch foliations.

Constructed Hirsch manifolds using exchangeable braided links.

Finitely many Hirsch manifolds exist for each strand number n.

Abstract

This paper is devoted to discussing affine Hirsch foliations on $3$-manifolds. First, we prove that up to isotopic leaf-conjugacy, every closed orientable $3$-manifold $M$ admits $0$, $1$ or $2$ affine Hirsch foliations. Furthermore, every case is possible. Then, we analyze the $3$-manifolds admitting two affine Hirsch foliations (abbreviated as Hirsch manifolds). On the one hand, we construct Hirsch manifolds by using exchangeable braided links (abbreviated as DEBL Hirsch manifolds); on the other hand, we show that every Hirsch manifold virtually is a DEBL Hirsch manifold. Finally, we show that for every $n\in \mathbb{N}$, there are only finitely many Hirsch manifolds with strand number $n$. Here the strand number of a Hirsch manifold $M$ is a positive integer defined by using strand numbers of braids.